190 research outputs found
Universal description of three two-component fermions
A quantum mechanical three-body problem for two identical fermions of mass
and a distinct particle of mass in the universal limit of zero-range
two-body interaction is studied. For the unambiguous formulation of the problem
in the interval ( and ) an additional parameter determining the wave function near
the triple-collision point is introduced; thus, a one-parameter family of
self-adjoint Hamiltonians is defined. The dependence of the bound-state
energies on and in the sector of angular momentum and parity is calculated and analysed with the aid of a simple model
On a class of second-order PDEs admitting partner symmetries
Recently we have demonstrated how to use partner symmetries for obtaining
noninvariant solutions of heavenly equations of Plebanski that govern heavenly
gravitational metrics. In this paper, we present a class of scalar second-order
PDEs with four variables, that possess partner symmetries and contain only
second derivatives of the unknown. We present a general form of such a PDE
together with recursion relations between partner symmetries. This general PDE
is transformed to several simplest canonical forms containing the two heavenly
equations of Plebanski among them and two other nonlinear equations which we
call mixed heavenly equation and asymmetric heavenly equation. On an example of
the mixed heavenly equation, we show how to use partner symmetries for
obtaining noninvariant solutions of PDEs by a lift from invariant solutions.
Finally, we present Ricci-flat self-dual metrics governed by solutions of the
mixed heavenly equation and its Legendre transform.Comment: LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions
of the Legendre transformed mixed heavenly equation and Ricci-flat metrics
governed by solutions of this equation are added. Eq. (6.10) on p. 14 is
correcte
Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere
equation so that this equation itself emerges as an algebraic consequence. We
regard the function in the extended Lax equations as a complex potential. We
identify the real and imaginary parts of the potential, which we call partner
symmetries, with the translational and dilatational symmetry characteristics
respectively. Then we choose the dilatational symmetry characteristic as the
new unknown replacing the K\"ahler potential which directly leads to a Legendre
transformation and to a set of linear equations satisfied by a single real
potential. This enables us to construct non-invariant solutions of the Legendre
transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler
metrics with anti-self-dual Riemann curvature 2-form that admit no Killing
vectors.Comment: submitted to J. Phys.
Effective three-body interactions in the alpha-cluster model for the ^{12}C nucleus
Properties of the lowest states of are calculated
to study the role of three-body interactions in the -cluster model. An
additional short-range part of the local three-body potential is introduced to
incorporate the effects beyond the -cluster model. There is enough
freedom in this potential to reproduce the experimental values of the
ground-state and excited-state energies and the ground-state root-mean-square
radius. The calculations reveal two principal choices of the two-body and
three-body potentials. Firstly, one can adjust the potentials to obtain the
width of the excited state and the monopole
transition matrix element in good agreement with the experimental data. In this
case, the three-body potential has strong short-range attraction supporting a
narrow resonance above the state, the excited-state wave function
contains a significant short-range component, and the excited-state
root-mean-square radius is comparable to that of the ground state. Next,
rejecting the solutions with an additional narrow resonance, one finds that the
excited-state width and the monopole transition matrix element are insensitive
to the choice of the potentials and both values exceed the experimental ones
Low-energy three-body dynamics in binary quantum gases
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose
mixtures is studied. Two identical fermions of the mass and a particle of
the mass with the zero-range two-body interaction in the states of the
total angular momentum L=1 are considered. Using the boundary condition model
for the s-wave interaction of different particles, both eigenvalue and
scattering problems are treated by solving hyper-radial equations, whose terms
are derived analytically. The dependencies of the three-body binding energies
on the mass ratio for the positive two-body scattering length are
calculated; it is shown that the ground and excited states arise at and ,
respectively. For m/m_1 \alt \lambda_1 and m/m_1 \alt \lambda_2, the
relevant bound states turn to narrow resonances, whose positions and widths are
calculated. The 2 + 1 elastic scattering and the three-body recombination near
the three-body threshold are studied and it is shown that a two-hump structure
in the mass-ratio dependencies of the cross sections is connected with arising
of the bound states.Comment: 16 page
Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?
Explicit Riemannian metrics with Euclidean signature and anti-self dual
curvature that do not admit any Killing vectors are presented. The metric and
the Riemann curvature scalars are homogenous functions of degree zero in a
single real potential and its derivatives. The solution for the potential is a
sum of exponential functions which suggests that for the choice of a suitable
domain of coordinates and parameters it can be the metric on a compact
manifold. Then, by the theorem of Hitchin, it could be a class of metrics on
, or on surfaces whose universal covering is .Comment: Misprints in eqs.(9-11) corrected. Submitted to Classical and Quantum
Gravit
Universal low-energy properties of three two-dimensional particles
Universal low-energy properties are studied for three identical bosons
confined in two dimensions. The short-range pair-wise interaction in the
low-energy limit is described by means of the boundary condition model. The
wave function is expanded in a set of eigenfunctions on the hypersphere and the
system of hyper-radial equations is used to obtain analytical and numerical
results. Within the framework of this method, exact analytical expressions are
derived for the eigenpotentials and the coupling terms of hyper-radial
equations. The derivation of the coupling terms is generally applicable to a
variety of three-body problems provided the interaction is described by the
boundary condition model. The asymptotic form of the total wave function at a
small and a large hyper-radius is studied and the universal logarithmic
dependence in the vicinity of the triple-collision point is
derived. Precise three-body binding energies and the scattering length
are calculated.Comment: 30 pages with 13 figure
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